Develop Skills and Strategies Lesson 11 Part 1: ntrouction CCLS ...
Develop Skills and Strategies
Lesson 11
Part 1: Introduction
CCLS
8.EE.B.5
Represent Proportional Relationships
You know how to identify proportional relationships and linear functions. Take a look
at this problem.
Only the top runners in the world can finish a marathon in close to 2 hours. This
graph shows the pace that one such runner followed in a race. What is the unit rate?
Marathon Pace
28
Distance (mi)
24
20
16
12
B
8
A
4
0
0
20
40
60 80 100 120 140
Time (min)
Explore It
Use math you already know to solve the problem.
Write the coordinates for points A and B.
How do you interpret these points in terms of the number of miles run in a given number
of minutes?
How can you use the information above to find the rate per minute?
Find the unit rate. Show your work.
What does the unit rate mean in the problem situation?
100
L11: Represent Proportional Relationships
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Part 1: Introduction
Lesson 11
Find Out More
The unit rate can be represented by the slope of a graph. Compare (0, 0) and (20, 4) in the
graph on the previous page. It shows that the rise is 4 and the run is 20. The slope, or ??rise??
run
¡¤¡¤¡¤
is ?? 4 ?? or ??1???.
20 ¡¤¡¤
5
¡¤¡¤
You can also use two points on the graph to find the slope. Use the formula below with
(0, 0) and (20, 4). Label one of the coordinates (x1, y1) and the other coordinate (x2, y2). Then
substitute the values for the variables in the formula for the slope.
?? y2 2 y1?? 5 ?? 4 2 0 ?? or ?? 4 ?? 5 ?? 1 ??
x2 2 x1
¡¤¡¤¡¤¡¤¡¤¡¤
20 2 0
¡¤¡¤¡¤¡¤¡¤¡¤
20
¡¤¡¤¡¤
5
¡¤¡¤
The y-intercept of the graph on the previous page is 0. The graphs of all proportional
relationships include the point (0, 0), so the y-intercept is always 0.
You have seen that equations of linear functions follow the format y 5 mx 1 b, where m is the
slope and b is the y-intercept. The graphs of proportional relationships also fit this format,
but the value of b is always 0. Let¡¯s look at the equation for the graph on the previous page.
slope
y 5 ??1??? x 1 0
y-intercept
5
¡¤¡¤
5 ?? 1?? x
5
¡¤¡¤
This means that all proportional relationships are linear functions in which the y-intercept is 0.
Reflect
1 Explain how it is possible that all proportional relationships are linear functions but not
all linear functions are proportional relationships.
L11: Represent Proportional Relationships
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101
Part 2: Modeled Instruction
Lesson 11
Read the problem below. Then explore different ways to represent and interpret the
unit rate of a proportional relationship.
A website contains hundreds of different recipes. Included with each recipe is a table
that gives the approximate cost for different numbers of servings. Find the unit cost
for this recipe. Then write an equation for finding the cost of x servings.
Roast Chicken with Vegetables and Rice
Number of Servings
2
4
6
5.00
Cost ($)
2.50
7.50
8
10.00
Picture It
Cost ($)
You can use a graph to represent a proportional relationship.
10
9
8
7
6
5
4
3
2
1
0
Recipe Cost
(1, 1.25)
0
1
2
3 4 5 6 7
Number of Servings
8
9
Model It
You can use an equation to represent the unit rate of a proportional relationship.
The graph shows that 1 serving of the recipe costs $1.25.
You can also divide any of the equivalent ratios in the table: ??2.50
?? 5 $1.25 and ??10?? 5 $1.25.
2
¡¤¡¤¡¤¡¤
8
¡¤¡¤
The equation for the cost of x servings is the unit rate 3 number of servings: y 5 1.25x.
102
L11: Represent Proportional Relationships
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Part 2: Guided Instruction
Lesson 11
Connect It
Now you will use the table, graph, and equation from the previous page to interpret
the unit rate.
2 Describe how the unit rate is shown in the table, graph, and equation.
3 What is the constant of proportionality? How does this relate to the unit rate?
4 What is the slope of the graph? How does this relate to the unit rate and constant of
proportionality?
5 Explain what the unit rate means in the context of the problem.
Try It
6 The table shows the function of how many words Gary can read if he reads at a constant
rate. Use the information in the table to make a graph, representing the number of
minutes on the horizontal axis. Find the slope of the graph and explain what it means in
this situation.
Number of Minutes
Number of Words
2
320
4
6
8
640 960 1,280
L11: Represent Proportional Relationships
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103
Part 3: Modeled Instruction
Lesson 11
Read the problem below. Then explore different ways to compare two different
proportional relationships.
The table and equation show the rates at which two different printers print in terms of
pages per minute. What is the difference between the per page rate of the two
printers? Which printer is faster?
Printer A
Number of Minutes
1
Number of Pages Printed
40
2
80
3
120
Printer B¡¯s printing
rate is defined by the
equation y 5 50x,
where x is the number
of minutes and y is
the number of pages
printed.
4
160
Picture It
You can use the equation for printer B to make a table of values.
Equation: y 5 50x
Printer B
Number of Minutes
1
Number of Pages Printed
50
2
100
3
150
4
200
Graph It
You can use the tables to graph the proportional relationships.
Printer Rates
Number of Pages Printed
160
140
120
Printer B
80
60
40
20
0
104
Printer A
100
0
1
2
3
4
Number of Minutes
5
L11: Represent Proportional Relationships
?Curriculum Associates, LLC
Copying is not permitted.
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